For our kernel, we will assume that such a \(\mathcal {G}_{\eta }\) -modulator is provided together with the input.
This has the technical reason that we otherwise cannot verify efficiently whether such a modulator actually exists.
In other words, we cannot determine efficiently for a graph \(G\)  with integers \(k\)  and \(\ell \)  whether there exists a feedback vertex set of size \(\ell \)  and a \(\mathcal {G}_{\eta }\) -modulator of size \(k\)  as the latter is an NP-hard problem.
By providing the modulator \(X\)  with the parameterized instance, we only need to verify that \(X\)  is indeed such a modulator.
This can be done efficiently when \(G-X\)  has bounded elimination distance to a forest, since determining elimination distances is fixed parameter tractable in the target value [1]}.
In practice, one can determine a \(\log (c+1)\) -approximation for the deletion distance to a minor-closed graph family with elimination distance to a forest bounded by \(c\)  [2]}, since the treewidth is bounded as well by Proposition REF .REF .
This approximation with an effectively constant approximation ratio improves the \({X_{\text{OPT}}} \log ^{3/2} {X_{\text{OPT}}}\) -approximation which was used in earlier papers [3]}, [4]}.
Here, \(X_{\text{OPT}}\)  denotes a minimum-sized modulator.
